### Homes

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

### Train Carriages

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

### Let's Investigate Triangles

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

# We'll Bang the Drum

## We'll Bang the Drum

Here is a pile of drums that will each give a beat.
Start off by placing two drums on the wheel. As the wheel turns, the drums will sound.
How many different rhythms can you make with just two drums?

Now try with 3 drums. As the wheel turns so the drums will play.
What different rhythms can you make now?
Are you sure they are all different from each other?

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Putting the drums in position and then turning the wheel can lead to some good discussions about whether there is a need to record, and if so how. Ideally, allow the children to come up with their own suggestions.

One way is in the form of a table putting the 6 places around the wheel across the table and marking each place where a drum beat would go.

This might lead to a table like this for three drums then four drums:

The children are likely to come up with more because of those that are actually equivalent to others. Again a lot of possibilities for discussion will arise about how they will decide which are the same and which are different. Is the first row actually the same as the fourth row in the table if the wheel spins for several turns? They can use the interactivity to listen to the rhythms carefully. The group will need to decide on their own "rules" about what is the same and what different.

This is a good opportunity to encourage children to find a system to work out the different combinations. For example, in the table above we have kept the first two drums in the same place and then found all the ways to put the third drum on the wheel. Then we have kept the first drum in the same place but altered the second drum's place and again found all the different positions for the third drum. Children might find this way of working (systematically) rather tricky, depending on their previous experience, so you will perhaps need to model a system for them in the first instance. Finding all the possibilities for three drums is quite challenging but some pupils could go on to look at four drums.

Another way to record might be more like how the wheel looks, in a pictorial representation. Here are some examples for three and four drums:

This way of recording might make it easier to spot which are the same and which different.